Extensions of the Kantorovich Inequality and the Bauer-Fike Inequality

Abstract

This paper proves a Kantorovich-type inequality on the matrix of the type $\frac{1}{2}(Q^H_1 AQ_1 Q^H_1A^{-1} Q_1+Q^H_1A^{-1}Q_1Q^H_1AQ_1)$, where $A$ is an $n\times n$ positive definite Hermitian matrix and $Q_1$ is an $n\times m$ matrix with rank $(Q_1)=m$. The result is applied to get an extension of the Bauer-Fike inequality on condition numbers of similarities that block diagonalized matrices.